Question

A small sphere of radius \(r\) is dropped in a viscous liquid. When it is moving with terminal velocity in the liquid, the relation between the rate of heat produced \[ \left(\frac{dQ}{dt}\right) \] and \(r\) is

• A. \(\dfrac{dQ}{dt}\propto r^2\)
• B. \(\dfrac{dQ}{dt}\propto r^5\)
• C. \(\dfrac{dQ}{dt}\propto \dfrac{1}{r^2}\)
• D. \(\dfrac{dQ}{dt}\propto \dfrac{1}{r^5}\) \bigskip

Hint:

For a sphere falling through a viscous liquid, \[ v_t\propto r^2. \] Since \[ F_{\text{viscous}}\propto r\,v_t, \] we get \[ F_{\text{viscous}}\propto r^3. \] Therefore, \[ \frac{dQ}{dt}=F_{\text{viscous}}v_t \propto r^5. \]

Answer 1

The Correct Option is B

Concept: Terminal velocity \(v_t \propto r^2\). Viscous force \(F \propto r v_t \propto r^3\). Heat rate \(\dot{Q} = F v_t \propto r^3 \cdot r^2 = r^5\).

Step 1: Write the final answer. \(\boxed{\frac{dQ}{dt}\propto r^5}\)